We solve problems

Problems Geometry Plane geometry Geometrical inequalities Triangle inequality

Show that if all sides of a triangle have integer lengths and one of them is equal to \(1\), then the other two have lengths equal to each other.

Problem #PRU-100680

Problems Geometry Triangle inequality

12–15 3.0

Two villages lie on the opposite sides of a river whose banks are straight lines. A bridge is to be built over the river perpendicular to the banks. Where should the bridge be built so that the path from one village to the other is as short as possible?

Problem #PRU-100681

Problems Geometry Triangle inequality

12–15 3.0

Quadrilateral \(ABCD\) is situated completely inside a quadrilateral \(EFGH\). Prove that the perimeter of \(ABCD\) is smaller than the perimeter of \(EFGH\).

Problem #PRU-100682

Problems Geometry Plane geometry Geometrical inequalities Triangle inequality

A billiard ball lies on a table in the shape of an acute angle. How should you hit the ball so that it returns to its starting location after hitting each of the two banks once? Is it always possible to do so?
(When the ball hits the bank, it bounces. The way it bounces is determined by the shortest path rule – if it begins at some point \(D\) and ends at some point \(D'\) after bouncing, the path it takes is the shortest possible path that includes the bounce.)

Problem #PRU-100683

Problems Geometry Triangle inequality

11–15 3.0

There are \(n\) mines and \(n\) cities scattered across the land, it is known that no three objects (mines, or cities) belong to one line. Every mine has to have a rail connection to exactly one city. Railways have to be straight and cannot cross other railways. Is it always possible?

Problem #PRU-100107

Problems Geometry Plane geometry Area Triangle area

11–13 4.0

In a parallelogram \(ABCD\), point \(E\) belongs to the side \(CD\) and point \(F\) belongs to the side \(BC\). Show that the total red area is the same as the total blue area:

Problem #PRU-100108

Problems Geometry Plane geometry Area Area of a polygon Finding the area by rearrangment

11–13 4.0

The figure below is a regular pentagram. What is larger, the black area or the blue area?

Problem #PRU-100109

Problems Geometry Plane geometry Area Area ratio Area ratio (other)

11–13 3.0

A circle was inscribed in a square, and another square was inscribed in the circle. Which area is larger, the blue or the orange one?

Problem #PRU-100110

Problems Geometry Plane geometry Area Finding the area by rearrangment

11–13 4.0

In a square, the midpoints of its sides were marked and some segments were drawn. There is another square formed in the centre. Find its area, if the side of the square has length \(10\).

Problem #PRU-100111

Problems Geometry Plane geometry Area Area (other) Finding the area by rearrangment

11–13 4.0

In a parallelogram \(ABCD\), point \(E\) belongs to the side \(AB\), point \(F\) belongs to the side \(CD\) and point \(G\) belongs to the side \(AD\). What is more, the marked red segments \(AE\) and \(CF\) have equal lengths. Prove that the total grey area is equal to the total black area.